When you double a bridge contract that is not game does it automatically double the contract into game?

Only two hearts or two spades will be doubled into 'game'.

The double just doubles the contract, it does not automatically put it into 'game'. If the bid was 'one', then it is doubled into 'two'.

If the bid was for 'two' it would double into 'four'. That is 'game' for hearts or spades. Four diamonds or four clubs would not be 'game'.

Doubling a two heart or two spade bid should be avoided unless you are very sure of defeating your opponents. If they make the two tricks, you have given them an easy 'game'.